Holomorphic automorphisms of compact Kähler surfaces and their induced actions in cohomology
Holomorphic automorphisms of compact Kähler surfaces and their induced actions in cohomology
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DOI：
10.1007/bf01403061
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发表时间：
1979-06
影响因子：
3.1
通讯作者：
C. Peters
中科院分区：
文献类型：
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作者：
C. Peters
对于任何紧复流形X，我们可以问X的全纯自同构群Aut (X)是否忠实作用于上同调环H*（X； A）上，其值在某个环A上。如果Aut (X)的恒等分量包含不同于1的元素g，则g在上同调上起平凡作用。所以答案是否定的如果t (X)的李代数不能约为{0}-或者等价地如果X允许一个非零全纯向量场。如果X与Y X IP'生物全纯同构，就会发生这种情况。现在，我们来看一下X是紧化黎曼曲面的情况。由于之前给出的原因，如果X的属是0或1，答案是负的。然而，一个著名的定理——回到Hurwitz——指出，在所有其他情况下，即如果属至少为2，群Aut (X)确实忠实地作用于ht （X, 7/）。看一下这个定理的证明是有益的，因为它包含了下面主要定理的一些成分。因此，假设X是> 2属的紧黎曼曲面，并且假设14 = geAutX对HI （X, 7/）起平凡作用。现在X上的正则系统没有基点，所以对于任何类型的X，存在一个在p处不消失的全纯l型e)。由于X上全纯1型的向量空间是HI （X, r）的直接因子，我们必须有g* o = eg。特别地，如果peX是g的一个不动点，那么在p点的余切空间上的诱导映射就是恒等。然后g= 1，与我们的假设相反。所以g没有不动点，Lefschetz不动点公式表明Trace g* l Hl (X, 7/)= 2。然而g*= id，所以Trace g* lH 1 (X, 7/)= rank h1 (X, 71)> 3，因为X的属至少是2。这个矛盾完成了证明。现在我们来看看紧复二维流形，它被称为曲面。为了完整起见，让我回顾一下在这种情况下所知道的情况。对于k3曲面X，群Aut (X)忠实地作用于HZ (X, 7/)(参见。Burns-Rapoport,[2], Prop. 1.1)和一个类似的命题对于Enriques曲面也是成立的（参见Ueno，[7]）。注意，在第一种情况下，H2 （X, 7/）没有扭转，而在第二种情况下，它有扭转。事实上，有一个叫恩里克的人
For any compact complex manifold X we may ask whether the group Aut (X) of holomorphic automorphisms of X acts faithfully on the cohomology ring H*(X; A) with values in some ring A. If the identity component of Aut (X) contains elements g different from 1 then g acts trivially in cohomology. So the answer is" no" if the Lie-algebra of Aut (X) doesn't reduce to {0}-or equivalently if X admits a non-zero holomorphic vectorfield. This happens if eg X is biholomorphically isomorphic to Y x IP'. Now, let me look at the case dimr ie X is a compact Riemann surface. Because of the reason given before, if the genus of X is 0 or 1 the answer is negative. However, a well-known theorem-going back to Hurwitz states that in all other cases, ie if the genus is at least 2, the group Aut (X) does operate faithfully on H t (X, 7/). It is instructive to look at the proof of this, since it contains some of the ingredients of the main theorem stated below. So, suppose X is a compact Riemann surface of genus> 2, and assume 1 4= geAutX acts trivially on HI (x, 7/). Now the canonical system on X is free of base points, so for any pe X there exists a holomorphic l-form e) which does not vanish at p. Since the vector space of holomorphic 1-forms on X is a direct factor of HI (X, r we must have that g* o)= eg. In particular, if peX were a fixed point of g, the induced map on the cotangent space at p would be the identity. But then g= 1, contrary to our assumptions. So g acts fixed point free, and the Lefschetz fixed point formula implies that Trace g* l Hl (X, 7/)= 2. However g*= id, so Trace g* lH 1 (X, 7/)= rank H 1 (X, 71)> 3, since the genus of X is at least 2. This contradiction completes the proof. Now we go over to the case of compact complex 2-dimensional manifolds, to be called surfaces. For the sake of completeness let me recall what is known in this situation.For K3-surfaces X the group Aut (X) operates faithfully on HZ (x, 7/)(cf. Burns-Rapoport,[2], Prop. 1.1) and a similar statement is true for Enriques surfaces (cf. Ueno,[7]). Notice that, whereas in the first case H2 (X, 7/) has no torsion, in the second case it does have torsion. In fact there exists an Enriques